The ring structure for equivariant twisted K-theory

被引:7
|
作者
Tu, Jean-Louis [1 ]
Xu, Ping [2 ]
机构
[1] Univ Paul Verlaine Metz, ISGMP, LMAM CNRS UMR 7122, F-57000 Metz, France
[2] Penn State Univ, Dept Math, University Pk, PA 16802 USA
来源
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK | 2009年 / 635卷
关键词
BAUM-CONNES CONJECTURE; LIE-GROUPS; GROUP COHOMOLOGY; MODULI SPACES; FOLIATIONS; GROUPOIDS; BUNDLES; STACKS; GERBES;
D O I
10.1515/CRELLE.2009.077
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an explicit construction of the transgression map T(1) : H* (Gamma(center dot); A) -> H*(-1) ((N x Gamma)(center dot);A) for any crossed module N -> Gamma and prove that any element in the image is infinity-multiplicative. As a consequence, we prove, under some mild conditions, for a crossed module N -> Gamma and any e is an element of Z(3) (Gamma(center dot);G(1)), that the equivariant twisted K-theory group K(e,Gamma)*(N) admits a ring structure. As an application, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted K-theory group K([e],G)*(G), defined as the K-theory group of a certain groupoid C*-algebra, is endowed with a canonical ring structure K([e],G)(i+d)(G) circle times K([e],G)(j+d)(G) -> K([e],G)(i+j+d)(G), where d = dim G and [c] is an element of H(2)((G x G)(center dot); S(1)). The relation with Freed-Hopkins-Teleman theorem [25] still needs to be explored.
引用
收藏
页码:97 / 148
页数:52
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